## Trace-penalty minimization for large-scale eigenspace computation.(English)Zbl 1373.65026

The eigenspace corresponding to the $$k$$ smallest eigenvalues of a large-scale matrix pencil $$(A,B)$$ with symmetric matrices of size $$n\times n$$ (and $$B$$ positive definite) is the solution of the constrained problem $$\min_X \mathrm{tr}(X^TAX)$$ with $$X\in\mathbb{R}^{n\times k}$$ satisfying $$X^TBX=I$$. It is proposed to solve this problem by minimizing $$f_\mu(X)=\frac{1}{2}\mathrm{tr}(X^TAX)+\frac{\mu}{4}\|X^TBX-I\|_F$$ using standard methods. A key result shown is that $$\mu$$ need not be very large, but that if suffices to choose $$\max(0,\lambda_k)<\mu<\lambda_n$$ where $$\lambda_1\leq\lambda_2\leq\cdots\leq\lambda_n$$ are the sorted eigenvalues of $$A$$ for there to be essentially only one minimum that is global. Algorithmic aspects of a classical steepest descent minimization procedure are discussed. This includes restarts, the choice of the parameter $$\mu$$, and stopping criteria. Also the complexity of the algorithm is discussed and it is extensively tested on numerical examples.

### MSC:

 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 65K05 Numerical mathematical programming methods 90C06 Large-scale problems in mathematical programming 15A22 Matrix pencils 65Y20 Complexity and performance of numerical algorithms

### Software:

lobpcg.m; ARPACK; LAPACK; PARSEC; BLOPEX; PRIMME; ScaLAPACK; KSSOLV
Full Text:

### References:

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